We note that we were not able to reduce the gradient norm to the requested tolerance of \(10^{-8}\) . This result can be attributed to approximations made in computing the gradient with the adjoint method. As discussed in [1]}, [2]}, the adjoint equations are derived under the assumption that the VMEC code satisfies MHD force balance (). However, there is always some residual error in force balance due to discretization error and the assumption of continuously nested flux surfaces. Furthermore, the linear adjoint solution is approximated by adding a small perturbation to the nonlinear MHD force balance. This technique is effectively a forward-difference approximation of the adjoint equation, which introduces an error that scales with the magnitude of the perturbation. When there are small errors in the gradient, the computed gradient may no longer be a descent direction, making convergence difficult near the optimum [3]}.
